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How to construct a Hovey triple from two cotorsion pairs

Volume 230 / 2015

James Gillespie Fundamenta Mathematicae 230 (2015), 281-289 MSC: Primary 55U35; Secondary 18G25, 18E10. DOI: 10.4064/fm230-3-4


Let $\mathcal{A}$ be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs $(\mathcal{Q}, \widetilde{\mathcal{R}})$ and $(\widetilde{\mathcal{Q}}, \mathcal{R})$ in $\mathcal{A}$ satisfying $\widetilde{\mathcal{R}} \subseteq \mathcal{R}$ and $\mathcal{Q} \cap \widetilde{\mathcal{R}} = \widetilde{\mathcal{Q}} \cap \mathcal{R}$. We show how to construct a (necessarily unique) abelian model structure on $\mathcal{A}$ with $\mathcal{Q}$ (resp. $\widetilde{\mathcal{Q}}$) as the class of cofibrant (resp. trivially cofibrant) objects, and $\mathcal{R}$ (resp. $\widetilde{\mathcal{R}}$) as the class of fibrant (resp. trivially fibrant) objects.


  • James GillespieRamapo College of New Jersey
    School of Theoretical and Applied Science
    505 Ramapo Valley Road
    Mahwah, NJ 07430, U.S.A.

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